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Angular distribution of scattered electrons associated with collimated bremsstrahlung and the tagging technique
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Angular distribution of scattered electrons associated with collimated bremsstrahlung and the tagging technique L.C. Maximon a,, J. Ahrens b, M. Dugger c a b c
Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, DC 20052, USA ¨t Mainz, D55099 Mainz, Germany ¨ r Kernphysik, Universita Institut fu Arizona State University, Tempe, AZ 85287, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 January 2009 Accepted 4 February 2009 Available online 20 February 2009
We investigate the angular correlation between a bremsstrahlung photon and its corresponding postbremsstrahlung electron within the context of a magnetic tagging spectrometer with the aim of improving the instrument’s efficiency. Our results are given in terms of angular distributions of the post-bremsstrahlung electron associated with photons that pass through a circular collimator centered in the forward direction. We start from the fully differential Bethe–Heitler (first Born approximation) cross-section, including the Molie`re screening correction, which is then integrated over the photon azimuthal angle and over the photon polar angle defined by the collimator. These integrations are performed analytically, making no high energy or small angle approximations. To obtain numerical values from the results of these integrations a multiprecision program is used to avoid severe problems of cancellations, especially at high energies (above 1 GeV). Making use of the angular correlation between the electron and the photon, we show that it is possible to increase the usable photon flux if we avoid the detection of electrons with large angles that have no photon partner passing the collimator. This can be accomplished by limiting the size of the electron detectors in the plane perpendicular to the bend plane. & 2009 Elsevier B.V. All rights reserved.
Keywords: Bethe–Heitler cross-section Molie`re screening Electron–photon correlation Collimated bremsstrahlung Tagging spectrometer Tagging efficiency Multiprecision Fortran program
1. Introduction Bremsstrahlung photons have been an important tool for investigating photonuclear reactions. A coherent summary of cross-section formulas and related data was published long ago [1]. This has been very useful, but with the advent of high duty factor electron accelerators, electron-bremsstrahlung tagging has become important for the production of quasi-monochromatic photons and the experimental determination of the photon flux. For this technique it is necessary to go beyond the formulas given in Ref. [1] since now the photons get collimated. We started from the fully differential Bethe–Heitler cross-section as given in formula 1BS in Ref. [1] and used the expression of Molie`re [2] for the atomic form factor in order to study the angular correlation between the post-bremsstrahlung electron and the photon. For small momentum transfers, the bremsstrahlung cross-section is large and the screening correction substantial. Since the photon collimator generally has a round bore in the forward direction, we are interested in the cross-section integrated analytically over the photon azimuth. In addition to the integration in the azimuthal direction, the photon polar angle is integrated analytically. To obtain numerical values from the results of these integrations a multiprecision program is needed to avoid severe
Corresponding author.
E-mail address: [email protected] (L.C. Maximon). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.02.007
problems of cancellations, especially at high energies (above 1 GeV). For this work we use the Fortran 90 libraries MPFUN90 of Bailey [3]. We discuss the properties of bremsstrahlung and the basic functioning of a bremsstrahlung tagger and conclude with a discussion of efficiencies arising from such an instrument. 2. Symbols and units In what follows we use the units given in the review article of Koch and Motz [1].
1 ; 2 ¼ initial; final electron total energy in mc2 units p1 ; p2 ¼ initial; final electron momentum in mc units k; k ¼ energy; momentum of the emitted photon; in mc2 and mc units q ¼ momentum transferred to the nucleus; in mc units e2 =_c ¼ 1=137:04 e2 =mc2 ¼ classical electron radius; r 0 m ¼ electron mass
y1 ¼ ffðk; p1 Þ
w ¼ ffðp2 ; p1 Þ y1c ¼ 1=1 photon characteristic angle wc ¼ y1c k=2 electron characteristic angle (1) We then have 2 p2 ¼ 1.
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3. General properties of bremsstrahlung and the functioning of a bremsstrahlung tagger In the electron-bremsstrahlung process, an electron with energy 1, when passing through the Coulomb field of a nucleus, radiates a photon with energy k. This energy k can take any value up to the kinetic energy of the electron. Since the photon and scattered electron are emitted in the forward direction and the mass of the electron is small compared to that of the nucleus, the momentum transferred to the nucleus is small and the recoil energy given to the nucleus can be neglected. The energy loss of the electron corresponds, therefore, to the energy of the bremsstrahlung photon. In the tagging technique, the energy of the post-bremsstrahlung electron, 2 , is measured in a wide band magnetic spectrometer, the tagger, and the energy of the tagged photon can be determined as k ¼ 1 2 . Typically the tagger is provided with a well granulated focal plane detector, such that 2 can be measured with the desired precision. In order to have a beam spot with a defined, limited size, the photons have to be collimated before they hit the target of interest. The number of electrons detected in the spectrometer is always larger than the number of coincident photons that hit the target under investigation; this is due not only to the photon collimation but also to the fact that electrons from radiationless reactions are detected in the spectrometer (e.g., Møller scattering). However, the number of electrons is always proportional to the number of tagged photons that pass the collimator. The ratio of the number of photons coincident with an electron (the tagged photons) to the total number of detected electrons is called the tagging efficiency Z. This quantity can easily be measured, e.g.. when, instead of the target of interest, a photon detector with known efficiency is installed in the photon beam and the tagged photons are scaled in coincidence with the number of electrons. Knowing Z one obtains the photon flux on the target of interest from the flux of detected electrons. A typical value of the tagging efficiency, 50%, is found when the photon collimation allows only photons with emission angles smaller than 1 characteristic angle, y1c ¼ 1=1 , to pass the collimator and all electrons are detected. The tagged photon flux is, in this technique, limited by the fact that for all tagged photons, the corresponding electrons have to be detected independently of whether the photon will cause a reaction in the target of interest or not. In order to increase the possible flux of photons, the granulation of the focal plane electron detector can be enhanced but there are limits to this technique, since the counting capabilities of single electron detectors have to be high. Another way to increase the usable photon flux is to avoid detecting electrons that have no photon partner passing the collimator. This can be done by making use of the angular correlation between the electron and the photon. Since the momentum transferred to the nucleus in the bremsstrahlung process is small, a substantial correlation is to be expected. The bremsstrahlung cross-section is large for small momentum transfers, q. The momentum transferred to the nucleus in the process of interest is an important parameter that influences the electron–photon angular correlation. Therefore, in our calculations screening corrections have to be taken into account since they decrease the bremsstrahlung cross-section significantly in the region of small momentum transfers. Since this is the only place where Z plays a role for the features discussed here, we always show cross-sections divided by Z 2. Bremsstrahlung crosssections scale well when polar angles are expressed in characteristic angles for the photon and the post-bremsstrahlung electron. So, in the figures that follow, the angles y1 and w are given in units of the respective characteristic angles.
radiator
electron detector
Fig. 1. The figure shows the optics of a typical tagger. Electrons with different polar angles start from the radiator. The dotted lines are electrons with polar angles in the bend plane, in which there is point to point focusing. Therefore, the main trajectory (solid line) and the dotted lines meet at the electron detector. The dashed lines show electrons with polar angles perpendicular to the bend plane. As shown in the figure, there is no focusing for these electrons. Thus, when they have large enough polar angles they miss the electron detector. If the bend plane is coincident with the x-plane, then the x-component of the electron polar angle gets focused, while electrons with a y-component larger than a corresponding width of the electron detector, miss this detector, and can said to be ‘‘collimated’’.
Tagging spectrometers are always built to have a good momentum resolution for the post-bremsstrahlung electron. This requires that in the bend plane point to point optics are realized, which leads to an integration over the angular distribution of the electrons in this plane. In the plane perpendicular to the bend plane one has, however, the chance to avoid the detection of electrons with large angles, for example by limiting the size of the electron detectors in this direction. The general optics of a tagging spectrometer are shown in Fig. 1.
4. The Bethe–Heitler cross-section In this section, we give the principal equations in the derivation of the angular distribution of scattered electrons associated with collimated bremsstrahlung from high energy electrons. The starting point for our derivation is the Bethe–Heitler (first Born approximation) cross-section. We are primarily interested in small angles between the momenta of the incident electron, the scattered electron, and the emitted photon, p1 ; p2 , and k, respectively, in which case the momentum transfer, q, is of order mc ¼ 1 or smaller. We, therefore, neglect the effects of nuclear recoil but include the effects of atomic screening. However, we make no high energy or small angle approximations in the analysis of the Bethe–Heitler cross-section. As generally written, the Bethe–Heitler cross-section [1], is expressed in a coordinate system in which the z-axis is in the direction of the emitted photon, k. In this system, the direction of the incident beam is given by the angles ðy1 ; f1 Þ and the direction of the scattered electron is given by ðy2 ; f2 Þ, where
y1 ¼ ffðp1 ; kÞ y2 ¼ ffðp2 ; kÞ
(2)
The azimuthal angles f1 and f2 are in the plane perpendicular to k. With the experimental setup in mind, we choose a coordinate system with the z-axis in the direction of the incident electron beam, p1 , rather than the direction of the emitted photon, We, therefore, express the cross-section in terms of the angles w; jp for the scattered electron, and y1 ; jk for the photon. Here
w ¼ ffðp2 ; p1 Þ y1 ¼ ffðk; p1 Þ
(3)
and the azimuthal angles jp and jk are in the plane perpendicular to p1 .
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The needed relations between these variables in the two coordinate systems are cos y2 ¼ cos y1 cos w þ sin y1 sin w cosðjk jp Þ
(4)
and cos w ¼ cos y1 cos y2 þ sin y1 sin y2 cosðf1 f2 Þ
(5)
from which sin y1 sin y2 cosðf1 f2 Þ ¼ cos wsin2 y1 cos y1 sin y1 sin w cosðjk jp Þ
(6)
The momentum transfer is related to the other variables by
with
a1 ¼ 0:10;
a2 ¼ 0:55;
bi ¼ ðZ 1=3 =121Þbi ;
a3 ¼ 0:35
b1 ¼ 6:0;
b2 ¼ 1:20;
b3 ¼ 0:30
(16)
Figs. 2 and 3 show correlations between the photon and the electron polar angles, y1 and w, for 1 ¼ 12 GeV and k ¼ 6 GeV and a difference in the azimuth of p. The sharp dips when the polar angles in units of the respective characteristic angles are equal result from phase space (see also Figs. 5–7 in Ref. [4]).
5. Integration over the azimuth
q ¼ ðp1 p2 kÞ2 2
2
2
¼ ðp1 p2 Þ2 k þ 2k 2kðp1 p2 Þ 2
¼ 2kd1 2kd2 þ m
(7)
where d1 ¼ ð1 p1 cos y1 Þ d2 ¼ ð2 p2 cos y2 Þ
(8)
m2 ðp1 p2 Þ2 k2 ¼ 2ð1 2 p1 p2 cos w 1Þ m20 ðp1 p2 Þ2 k2 ¼ 2ð1 2 p1 p2 1Þ
(9)
We start with the totally differential Bethe–Heitler crosssection for the scattering of an electron with the emission of a photon 2 e2 e2 ½1 FðqÞ2 p2 dk dOk dOp2 3 f gBH d s ¼ Z2 _c mc2 p1 k ð2pÞ2 q4
(10)
where f gBH ¼
With the integration over jk in mind, we note from Eqs. (4) and (7) that in the above expression for the cross-section the variables that depend on jk are d2 and q2 . Writing f ¼ jk jp , the integral over the photon azimuthal angle is then, from Eqs. (13) and (15), 2 e2 e2 p2 dk sin y1 dy1 dOp2 3 d s ¼ Z2 2 _c mc p1 k ð2pÞ2 9 8 Z 2p Z 2p 3 = < X X ai aj 2 2 ½ ij df þ ai ½ i df 2 2 ; : 0 ðb b Þ 0 1piojp3
in which " ½ ij ¼
2
2 2
2
sin y1 ð4 q Þ
ð1 p1 cos y1 Þ2
þ
p22
2
2 1
2
sin y2 ð4 q Þ
2
2
q2 þ bi
2
q2 þ bj
1 2
ðq2 þ bi Þ2
(20)
2
1 = 12 GeV, k = 6 GeV, k - p =
p1 p2 sin y1 sin y2 cosðf1 f2 Þ ¼ p1 p2 cos w ð1 d1 Þð2 d2 Þ (12) The expression f gBH in Eq. (11) may then be written as ( 2 1 2 2ð21 þ 22 Þ m2 2 f gBH ¼ ð2kÞ2 4 þ q 1
D2
D1 D2
1 1 1 1 1 þ q2 þ ðq2 m2 Þ D1 D2 D1 D2 2
2 )
(13)
where
5 2
1/c = 0.0
Z=6
1/c = 0.5 1/c = 1.0
12
d3 /(Z2 dk dΩk dΩp2)/(barns/(sr2 mc2))
Eqs. (4), (6) and (7) then express all of the terms in Eq. (11) in the variables of the p1 system. In view of the denominators ð1 p1 cos y1 Þ and ð2 p2 cos y2 Þ, we write Eq. (11) in the variables given in Eqs. (8) and (9). From Eq. (5) we then have
10
5
1/c = 1.5 Z = 79
2 1011 5 2 1010 5 2 109
(14)
For the atomic form factor in Eq. (10) we use the expression given by Molie`re [2], viz., 3 1 FðqÞ X ai ¼ 2 2 q2 i¼1 q þ bi
(19)
D2 ¼ 2kd2 ¼ c þ d cos f
(11)
D1 ¼ 2kd1 D2 ¼ 2kd2
(18)
f gBH
f gBH
2k ðp21 sin y1 þ p22 sin y2 2p1 p2 sin y1 sin y2 cosðf1 f2 ÞÞ ð1 p1 cos y1 Þð2 p2 cos y2 Þ
D1
#
1
From Eq. (4) we have
ð2 p2 cos y2 Þ2
2p p sin y1 sin y2 cosðf1 f2 Þð41 2 q2 Þ 1 2 ð1 p1 cos y1 Þð2 p2 cos y2 Þ þ
2
(17)
i¼1
i
and ½ i ¼
p21
1
j
(15)
5 -1.0
-0.5
0.0
0.5 /c
1.0
1.5
2.0
Fig. 2. The Bethe–Heitler cross-section with screening correction as a function of the electron polar angle for different fixed photon polar angles and different values of Z.
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where (
2
422 þ bi
X i ¼ ð2kÞ2
D21
!
2
I10;i 2
41 2 þ bi
D1
2
I11;i þ ð421 þ bi ÞI12;i
1 1 2 I10;i ðm2 þ bi Þ I10;i I11;i D1 2 2 2 2 2 2ð1 þ 2 Þ m I11;i bi
(26)
D1
d3
(
2
422 þ bi
Y i ¼ ð2kÞ2
D21
1
c
1.1
2
3
1
4
5
grees
p / de
Fig. 3. The Bethe–Heitler cross-section with screening correction as a function of electron polar angle and the electron azimuth. The photon polar angle is y1c . Note that the line for jp ¼ 0 corresponds to the curve in Fig. 2 for y1 =y1c ¼ 1 and Z ¼ 6.
in which
2 e2 e2 p2 dk sin y1 dy1 dOp2 3 d s ¼ Z2 _c mc2 p1 k ð2pÞ2 9 8 = < 3 X X ai aj 2 ½X 2 X þ a Y i j i i 2 2 ; : ðb b Þ
c ¼ 2kð2 p2 cos y1 cos wÞ (21)
c d ¼ 2kð2 p2 cosðy1 wÞÞ40
I01 ¼
The variables ai and b are defined by
I20;i
2
q2 þ bi ¼ ai þ b cos f
(22) I02
from which, using Eqs. (4) and (7), 2
ai ¼ 2kd1 þ m2 2kð2 p2 cos y1 cos wÞ þ bi b ¼ 2kp2 sin y1 sin w ¼ d
I11;i (23)
from which, with some algebra, I21;i
2
ai b ¼ d þ bi þ 4dk sin2 12 y1 þ 4dp2 sin2 12w þ 4ðp2 þ kÞ2 sin2 12 y1 sin2 12 w þ 4ðp2 sin 12 w cos 12y1 k cos 12w sin 12y1 Þ2 40
I12;i
The integrals over f are, thus, all of the form df
0
ðq2 þ bi Þm Dn2
Z 2p ¼ 0
2
df ðai þ b cos fÞm ðc þ d cos fÞn
I22;i m; n ¼ 0; 1; 2
(24)
From Eqs. (18)–(24) the integrals over the photon azimuthal angle in Eq. (17) can then be written in the form
0
0
½ ij df ¼ X i X j ½ i df ¼ Y i
i
(28)
i¼1
I00 ¼ 2p I10;i ¼
from which
Z 2p
j
All of the integrals Imn;i can be calculated recursively from Im0;i and I0n . From Eq. (24) we have
d ¼ 2kp2 sin y1 sin w
2
(27)
The cross-section integrated over the photon azimuthal angle, given in Eq. (17), is thus expressed in terms of X i and Y i by Eqs. (25)–(27) in terms of the integrals Imn;i defined in Eq. (24), specifically,
1piojp3
Z 2p
D1
2
I21;i þ ð421 þ bi ÞI22;i
D1
D1
/
Z 2p
41 2 þ bi
2 2ð1 þ 22 Þ m2 2 I20;i þ I11;i bi I21;i D1 1 1 1 1 2 I20;i I21;i þ I10;i I11;i ðm2 þ bi Þ D1 2 2 D1 !) 1 2 þ I I11;i þ I12;i 2 10;i
0 0.9
Imn;i
!
2
I20;i 2
(25)
2p ða2i
2
b Þ1=2 2p 2
ðc2 d Þ1=2 @ 2pai ¼ I ¼ @ai 10;i ða2 b2 Þ3=2 i @ 2pc ¼ ; I01 ¼ 2 @c ðc2 d Þ3=2 Z 2p 1 1 1 df ¼ þ c þ ai 0 ai þ b cos f c þ d cos f 1 ¼ ðI þ I01 Þ c þ ai 10;i @ ¼ I @ai 11;i 1 ¼ ðI þ I11;i Þ c þ ai 20;i @ ¼ I @c 11;i 1 ¼ ðI þ I02 Þ c þ ai 11;i @ ¼ I @ai 12;i 1 ¼ ðI þ I12;i Þ c þ ai 21;i
(29)
Here 2
c þ ai ¼ D1 þ m2 þ bi G1;i
(30)
The variables needed for the computation of the integrals Imn;i 2 2 are ai ; c; c þ ai ; a2i b , and c2 d . We next integrate the crosssection over the photon polar angle corresponding to the collimator. However, for the integration we choose the variable
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D1 rather than y1 , defined in Eqs. (8) and (14) by D1 ¼ 2kd1 ¼ 2kð1 p1 cos y1 Þ, from which dD1 ¼ 2kp1 sin y1 dy1 . We integrate the cross-section over y1 from y1 ¼ 0 to y0 and note that for y1 ¼ 0 we have D1 ¼ D1;0 ¼ 2kð1 p1 Þ ¼ 2k=ð1 þ p1 Þ and for y1 ¼ y0 we have D1 ¼ D1;1 ¼ 2kð1 p1 cos y0 Þ ¼ 2k=ð1 þ p1 Þþ 2kp1 ð1 cos y0 Þ ¼ 2k=ð1 þ p1 Þ þ 4kp1 sin2 12y0 . We then have 2 e2 e2 p2 dk dOp2 1 _c mc2 p1 k ð2pÞ2 2p1 k 8 Z D1;1 < X ai aj 2 ½X i X j dD1 2 2 : 1piojp3 ðbj bi Þ D1;0 ) Z D1;1 3 X 2 þ ai Y i dD1
2
Z
D1;1
D1;0
Y i dD1 ¼ ð2kÞ2
(31)
D1;0
2
2
þ ð2ð21 þ 22 Þ m2 ÞðJ 1;11;i bi J1;21;i Þ 1 2 ðm2 þ bi ÞðJ1;20;i J0;21;i Þ 2 1 þ ðJ 1;10;i J 0;11;i Þ þ ðJ2;10;i 2J1;11;i þ J 0;12;i Þ 2
D1;0
Imn;i
Dl1
dD1
(32)
D1;1 D1;0
2
X i dD1 ¼ ð2kÞ
l ¼ 0; 1
2 2
ðð4 þ
b2i ÞJ 2;10;i
2ð41 2 þ
(35)
J0;2n;i
n ¼ 0; 1; 2
J1;2n;i
n ¼ 0; 1
I12;i ¼
2
I22;i ¼
2
bi ð2ð21 þ 22 Þ m2 ÞJ 1;11;i
(33)
(36)
Using the recursion relations (29) for Imn;i, we need only to calculate explicitly the integrals for which either m or n is zero, the remaining integrals then being expressed in terms of Im0;i and I0n;i . To this end, we first note, from Eqs. (29) and (30) that
I21;i ¼
b2i ÞJ 1;11;i
þ ð421 þ bi ÞJ 0;12;i Þ J 0;10;i 1 2 ðm2 þ bi ÞðJ 1;10;i J 0;11;i Þ 2
1
G1;i 1
G1;i 1
G1;i 1
G21;i
ðI10;i þ I01 Þ I20;i þ I02;i þ
1 = 12 GeV, k = 6 GeV Z=6
1/1c = 0.5
d3 /(Z2 dk d1 dΩp2)/(barns/(rad sr mc2))
10
13
5
Z = 79
ðI10;i þ I01 Þ 2
G31;i
ðI10;i þ I01 Þ
Defining Z D1;1 Imn;i K l;mn;i dD1 l
G1;i
1/1c = 1.0
J0;11;i ¼ K 1;10;i þ K 1;01;i
1/1c = 1.5
J0;12;i ¼ K 1;02;i þ K 2;10;i þ K 2;01;i
(37)
(38)
J0;21;i ¼ K 1;20;i þ K 2;10;i þ K 2;01;i
2
J0;22;i ¼ K 2;20;i þ K 2;02;i þ 2K 3;10;i þ 2K 3;01;i 1 J1;11;i ¼ ðJ 1;10;i þ J 1;01 K 1;10;i K 1;01;i Þ m2 þ b2i 1 ðJ 1;02 K 1;02;i K 2;10;i K 2;01;i Þ J1;12;i ¼ m2 þ b2i 1 ðJ 1;10;i þ J1;01 K 1;10;i K 1;01;i Þ þ 2 ðm2 þ bi Þ2 1 ðJ 1;20;i K 1;20;i K 2;10;i K 2;01;i Þ J1;21;i ¼ m2 þ b2i 1 ðJ 1;10;i þ J1;01 K 1;10;i K 1;01;i Þ þ 2 ðm2 þ bi Þ2
1012 5 2 1011 5 2 1010 5 2 109 5 -1.0
1
G21;i
ðI10;i þ I01 Þ
we can then express all of the integrals in Eqs. (33) and (34) in terms of integrals J l;mn;i and K l;mn;i in which either m or n is zero. From Eqs. (32), (37) and (38) we have
1/1c = 0.0
2
1
G21;i
ðI20;i þ I02 Þ þ
D1;0
5
(34)
and the six in which m ¼ 2, namely
I11;i ¼
From Eqs. (26) and (27) Z
l ¼ 0; 1; 2
J2;20;i
Substituting Eqs. (26) and (27) in Eq. (28), the integrals over the photon polar angle are all of the form D1;1
Jl;10;i ; Jl;11;i ; J0;12;i
6. Integration over the photon polar angle
Z
2
As can be seen in the above expressions, the needed integrals are the six in which m ¼ 1, namely
In Fig. 4 a resulting cross-section is shown, which corresponds to Fig. 2 before integration.
J l;mn;i
2
ðð422 þ bi ÞJ 2;20;i 2ð41 2 þ bi ÞJ1;21;i
þ ð421 þ bi ÞJ 0;22;i Þ J0;20;i
d s ¼ Z2
i¼1
-0.5
0.0
0.5 / c
1.0
1.5
2.0
Fig. 4. The Bethe–Heitler cross-section with screening correction integrated over the photon azimuth jk .
(39)
The required integrals are now Jl;10;i ;
l ¼ 0; 1; 2
Jl;20;i ;
l ¼ 0; 1; 2
J1;01
(40)
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
(
and K l;10;i ;
l ¼ 1; 2; 3
K l;01;i ;
l ¼ 1; 2; 3
K l;20;i ; K l;02;i ;
l ¼ 1; 2 l ¼ 1; 2
) D1;1
B ð2pÞ2
2;i K 1;10;i
B3;i G1;i I10;i
B3;i D1;0 ( ) D1;1
D ð2pÞ2
¼ 2;i K 1;01;i
D3;i G1;i I01
D3;i
K 2;10;i ¼
K 2;01;i (41)
We note that although ai ; c and c þ ai are linear functions of 2 2 cos y1 and that a2i b and c2 d are quadratic functions of cos y1 , the expressions are simplified if we use D1 ¼ 2kd1 rather than cos y1 as the integration variable. In view of the integrands in 2 2 Eqs. (32) and (38) it is useful to write c2 d and a2i b alternatively as quadratic functions of D1 and of G1;i . We then have, with a bit of algebra,
ai ¼ g1 D1 þ g2;i ¼ g1 G1;i þ g3;i
K 3;10;i ¼
K 3;01;i ¼
J 2;20;i ¼
(42)
and 2
c2 d ¼ C 1 D21 þ 2C 2 D1 þ C 3 ¼ D1 G21;i þ 2D2;i G1;i þ D3;i 2
a2i b ¼ A1 D21 þ 2A2;i D1 þ A3;i ¼ B1 G21;i þ 2B2;i G1;i þ B3;i
(43)
K 2;20;i ¼
where from Eqs. (21) and (23) p2 cos w p1 2k 1 d2 ¼ 2 k þ 1 m2 2 p1
d1 ¼
d3;i ¼ d2 d1 ðm2 þ b2i Þ
2 p2 p1 2k 1 C 2 ¼ 2 k 2 m2 2 p1 2 2k 1 2 2 C3 ¼ m ðm þ 4Þ p1 4
K 2;02;i ¼
C1 ¼
2
D2;i ¼ C 2 C 1 ðm2 þ bi Þ 2
2
D3;i ¼ C 3 2C 2 ðm2 þ bi Þ þ C 1 ðm2 þ bi Þ2
(44)
and
g1 ¼ 1 d1
D1;0
þ
1 2 2 ðm þ k Þ p21 1 1 2 1 2 ¼ 2 m2 m k2 þ b2i m þ k1 2 2 p1 1 1 2 2 2 2 ¼ 2 ðm p2 bi p1 Þ þ 8bi p1 p2 ð1 2 þ p1 p2 1Þsin2 w 2 p1
A3;i
þ
B3;i
K 1;02;i ¼ þ
The integrals with l ¼ 2 or 3 can be expressed in terms of integrals for which l ¼ 0 or 1
K 1;10;i
D þ d1 d3;i 2;i D3;i D3;i
(45)
D1;0
g3;i
g3;i
("
2
) D1;1
A ð2pÞ2
¼ 2;i J1;10;i
A3;i D1 I10;i
A3;i D1;0 ( ) D1;1 ð2pÞ2
C2 ¼ J
C 3 D1 I01
C 3 1;01
(47)
1;0
B2;i ¼ A2;i A1 ðm2 þ bi Þ
(
J 1;10;i
!# ) D1;1
B B1 G1;i þ B2;i
I þ g1 g3;i 2;i 10;i
2
B3;i B3;i B1 B3;i B2;i D
K 1;20;i ¼
B1 ¼ A 1 B3;i ¼ D3;i
g2;i
("
A1 ¼
J 2;01
(46)
Furthermore, the integrals with l ¼ 1 and either m ¼ 2 or n ¼ 2 can be expressed in terms of the integrals with l ¼ 1 and either m ¼ 1 or n ¼ 1 (" !# ) D1;1
g2;i A A1 D1 þ A2;i
I þ g1 g2;i 2;i J1;20;i ¼
10;i 2
A3;i A3;i A1 A3;i A2;i
g2;i ¼ d2 þ ðm2 þ b2i Þ g3;i ¼ d3;i
J 2;10;i
) D1;1
3B ð2pÞ B1
2;i K 2;10;i K
2 2B3;i 2B3;i 1;10;i 2B3;i G1;i I10;i D 1;0 ( ) D1;1
3D2;i ð2pÞ2 D1
K K
2 2D3;i 2;01;i 2D3;i 1;01;i 2D3;i G1;i I01 D 1;0 0(" 1 # ) D1;1
A2;i ðA1 D1 þ A2;i Þ g1 @
I10;i
1 þ J1;10;i A
A3;i A1 A3;i A22;i D1;0 (" ! g2;i 3A2;i ðA1 D1 þ A2;i Þ 1 þ þ A3;i D1 A3;i A1 A3;i A22;i 1 !# )
D1;1 3A 3A22;i
2;i A I10;i
þ J 2A1
A3;i A3;i 1;10;i D1;0 0(" 1 # ) D1;1
B2;i ðB1 G1;i þ B2;i Þ g1 @
I10;i
1 þ K 1;10;i A
B3;i B1 B3;i B22;i D1;0 (" ! g3;i 3B2;i ðB1 G1;i þ B2;i Þ 1 þ þ B3;i G1;i B3;i B1 B3;i B22;i 1 !# ) D1;1
3B22;i 3B2;i
A 2B1 I10;i
þ K
B3;i B3;i 1;10;i D1;0 0(" 1 # ) D1;1
D2;i ðD1 G1;i þ D2;i Þ d1 @
I01
1 þ K 1;01;i A
D3;i D1 D3;i D22;i D1;0 (" ! d 3D2;i ðD1 G1;i þ D2;i Þ 1 þ þ 3;i D3;i G1;i D3;i D1 D3;i D22;i 1 !# )
D
1;1 3D 3D22;i
2;i A 2D1 I01
þ K
D3;i D3;i 1;01;i 2
D1;0
D1 ¼ C 1
A3;i
D1;0
(
c ¼ d1 D1 þ d2 ¼ d1 G1;i þ d3;i
A2;i
273
d3;i
d3;i D3;i
K 1;01;i
(48)
D1 G1;i þ D2;i D1 D3;i D22;i
!#
) D1;1
I01
D1;0
(49)
We now have only to evaluate explicitly the integrals for which l ¼ 0 or 1, in terms of which we have expressed all the other integrals. These are ( ) D1;1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
J0;10;i ¼ 2p pffiffiffiffiffiffi lnð ðA1 D1 þ A2;i Þ2 þ A1 A3;i A22;i þ ðA1 D1 þ A2;i ÞÞ
A1 D 1;0
(50)
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
(" J 0;20;i ¼
g1 ðA2;i D1 þ A3;i Þ þ g2;i ðA1 D1 þ A2;i Þ A1 A3;i
A22;i
#
) D1;1
I10;i
0<1/1c<1
D1;0
(52)
J 1;01
8 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 D1;1
< 1 ðC 2 D1 þ C 3 Þ2 þ ðC 1 C 3 C 22 ÞD21 þ ðC 2 D1 þ C 3 Þ =
A
¼ 2p pffiffiffiffiffiffi ln@ ;
: C3 D1 D1;0
(53)
K 1;10;i
0.009
D1;0
8 19 D1;1 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
< 1 ðA2;i D1 þ A3;i Þ2 þ ðA1 A3;i A22;i ÞD21 þ ðA2;i D1 þ A3;i Þ =
A
¼ 2p pffiffiffiffiffiffiffi ln@ : A3;i ;
D1
J1;10;i
1 = 12 GeV, k = 6 GeV, Z = 6, int 1
(51)
8 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 D1;1 < 1 ðB2 G1;i þ B3;i Þ2 þ ðB1 B3;i B22;i ÞG21;i þ ðB2 G1;i þ B3;i Þ =
A
¼ 2p pffiffiffiffiffiffiffi ln@ : B3;i ;
G1;i D1;0
1<1/1c<2
0.008
2<1/1c<3
d2 /(Z2 dk dΩp)/(barns/(sr mc2))
274
(54)
0.007
0<1/1c<10
0.006 0.005 0.004 0.003 0.002 0.001
8 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 D1;1 < 1 ðD2;i G1;i þ D3;i Þ2 þ ðD1 D3;i D22;i ÞG21;i þ ðD2 G1;i þ D3;i Þ =
A
K 1;01;i ¼ 2p pffiffiffiffiffiffiffiffi ln@ : D3;i ;
G1;i
0.0
D1;0
1
0
(55)
2
3 / c
Here 2
2 2
A1 A3;i A22;i ¼ B1 B3;i B22;i ¼ ½ðm2 þ bi Þ2 þ 4bi k
p22 sin2 w p21
(56)
4
5
6
Fig. 5. Angular distribution of post-bremsstrahlung electrons after integration over different intervals of the photon polar angle y1 .
and p22 sin2 w p21
0.009 (57)
Fig. 5 shows the angular distribution of the post-bremsstrahlung electrons as a function of the polar angle of the scattered electron for different photon collimations. The angular correlation can be seen clearly.
7. Final results and conclusion In order to investigate the angular correlation between a bremsstrahlung photon and its corresponding post-bremsstrahlung electron within the context of a tagging spectrometer, we integrated analytically the Bethe–Heitler (first Born approximation) cross-section, including the Molie`re screening correction. Our results are given in terms of angular distributions of the postbremsstrahlung electron associated with photons that pass through a circular collimator centered in the forward direction. To draw conclusions from this distribution we assume optical properties of the magnetic tagging spectrometer as indicated in Fig. 1. In the bend plane of the spectrometer, we have point to point optics in order to achieve good momentum resolution. This results in integration over the respective component of the electron polar angle. However, for the component perpendicular to the bend plane, we have the freedom to assume optics in which the electron polar angle results in a position that can be collimated by the size of the electron detectors. Integration over the angular distributions in Fig. 6 gives the cross-section for the different combinations of collimation, denoted by sij, with i and j representing the photon and the electron collimation, respectively, where 0 stands for off and 1 for on. We can then determine ratios of the si;j , which stand for the following efficiencies. The tagging efficiency Z is defined as the number of tagged photons through the collimator divided by the total number of detected electrons, collimated or uncollimated. For uncollimated electrons we find the efficiency Z0 s10 =s00. For the collimated
0.008
d2 /(Z2 dk dΩp)/(barns/(sr mc2))
C 1 C 3 C 22 ¼ D1 D3;i D22;i ¼ ð2kÞ2
0.007 0.006
1 coll
coll
off
off
on
off
off
on
on
on
0.005 0.004 0.003 0.002 0.001 0.0 0
1
2
3
4
5 /c
6
7
8
9
10
Fig. 6. Angular distribution of the electrons in coincidence with photons that pass the collimator, for different collimation of the photons and the electrons. There is either no collimation (off) or a collimation to 1 characteristic angle (on). All combinations are shown.
case we have Z s11 =s01 . The photon probability, b, is given by the ratio of tagged photons to all photons through the collimator, i.e. b s11 =s10 . Finally, we define l, which is the ratio of collimated electrons to uncollimated electrons, l s01 =s00 , which gives the reduction of electron count rate due to electron collimation. The above defined ratios are shown as a function of w=wc in Fig. 7. The tagging efficiency for uncollimated electrons, Z0 , is of course constant. Once the electrons are collimated, the tagging efficiency Z increases with decreasing number of detected
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
275
1.0 0.9 0.8
efficiencies
0.7 0.6 0.5
Z=6
Z = 79
0.4
1 coll = 0.5 1c
1 coll = 0.5 1c
0.3 0.2 0.1 0.0
0.9 0.8
efficiencies
0.7 0.6 0.5 0.4 0.3
Z=6
Z = 79
0
0.2
1 coll = 1.0 1c
1 coll = 1.0 1c
0.1
/
0.0 0.0
0.5
1.0 1.5 2.0 2.5 electron collimation in c
0.0
0.5
1.0 1.5 2.0 2.5 electron collimation in c
3.0
Fig. 7. Efficiencies deduced from the values of the sij for different photon collimations and Z. See text.
electrons, while the photon probability b decreases (since now we have photons through the collimator that are not tagged). In a tagged photon experiment, the number of ‘‘good’’ reactions is proportional to the number of tagged photons. When the postbremsstrahlung electrons are collimated, the photon probability b is no longer equal to one and there is a reduction by a factor of b in the number of tagged photons. To compensate for this reduction in tagged photons, the incoming electron current can be increased by a factor of 1=b so as to keep the number of ‘‘good’’ reactions the same as when there is no collimation on the post-bremsstrahlung electrons. When the post-bremsstrahlung electrons are collimated and the incoming electron current is increased by 1=b, the number of collimated electrons is decreased by a factor of l=b from the case where the post-bremsstrahlung electrons are not collimated, while the number of ‘‘good’’ events remains the same for each
case. Therefore, for a given good reaction rate, the tagger rate is decreased by a factor of l=b when the post-bremsstrahlung electrons are collimated.
Acknowledgment The work done at Arizona State University was supported by the National Science Foundation award PHY-0653630. References [1] [2] [3] [4]
H.W. Koch, J.W. Motz, Rev. Mod. Phys. 31 (1959) 920. G. Molie`re, Z. Naturforsch. 2a (1947) 133. D.H. Bailey, ACM Trans. Math. Software 19 (3) (1993) 288. L.C. Maximon, A. de Miniac, T. Aniel, Phys. Rep. 147 (1987) 189.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Angular distribution of scattered electrons associated with collimated bremsstrahlung and the tagging technique L.C. Maximon a,, J. Ahrens b, M. Dugger c a b c
Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, DC 20052, USA ¨t Mainz, D55099 Mainz, Germany ¨ r Kernphysik, Universita Institut fu Arizona State University, Tempe, AZ 85287, USA
a r t i c l e in f o
a b s t r a c t
Article history: Received 9 January 2009 Accepted 4 February 2009 Available online 20 February 2009
We investigate the angular correlation between a bremsstrahlung photon and its corresponding postbremsstrahlung electron within the context of a magnetic tagging spectrometer with the aim of improving the instrument’s efficiency. Our results are given in terms of angular distributions of the post-bremsstrahlung electron associated with photons that pass through a circular collimator centered in the forward direction. We start from the fully differential Bethe–Heitler (first Born approximation) cross-section, including the Molie`re screening correction, which is then integrated over the photon azimuthal angle and over the photon polar angle defined by the collimator. These integrations are performed analytically, making no high energy or small angle approximations. To obtain numerical values from the results of these integrations a multiprecision program is used to avoid severe problems of cancellations, especially at high energies (above 1 GeV). Making use of the angular correlation between the electron and the photon, we show that it is possible to increase the usable photon flux if we avoid the detection of electrons with large angles that have no photon partner passing the collimator. This can be accomplished by limiting the size of the electron detectors in the plane perpendicular to the bend plane. & 2009 Elsevier B.V. All rights reserved.
Keywords: Bethe–Heitler cross-section Molie`re screening Electron–photon correlation Collimated bremsstrahlung Tagging spectrometer Tagging efficiency Multiprecision Fortran program
1. Introduction Bremsstrahlung photons have been an important tool for investigating photonuclear reactions. A coherent summary of cross-section formulas and related data was published long ago [1]. This has been very useful, but with the advent of high duty factor electron accelerators, electron-bremsstrahlung tagging has become important for the production of quasi-monochromatic photons and the experimental determination of the photon flux. For this technique it is necessary to go beyond the formulas given in Ref. [1] since now the photons get collimated. We started from the fully differential Bethe–Heitler cross-section as given in formula 1BS in Ref. [1] and used the expression of Molie`re [2] for the atomic form factor in order to study the angular correlation between the post-bremsstrahlung electron and the photon. For small momentum transfers, the bremsstrahlung cross-section is large and the screening correction substantial. Since the photon collimator generally has a round bore in the forward direction, we are interested in the cross-section integrated analytically over the photon azimuth. In addition to the integration in the azimuthal direction, the photon polar angle is integrated analytically. To obtain numerical values from the results of these integrations a multiprecision program is needed to avoid severe
Corresponding author.
E-mail address: [email protected] (L.C. Maximon). 0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.02.007
problems of cancellations, especially at high energies (above 1 GeV). For this work we use the Fortran 90 libraries MPFUN90 of Bailey [3]. We discuss the properties of bremsstrahlung and the basic functioning of a bremsstrahlung tagger and conclude with a discussion of efficiencies arising from such an instrument. 2. Symbols and units In what follows we use the units given in the review article of Koch and Motz [1].
1 ; 2 ¼ initial; final electron total energy in mc2 units p1 ; p2 ¼ initial; final electron momentum in mc units k; k ¼ energy; momentum of the emitted photon; in mc2 and mc units q ¼ momentum transferred to the nucleus; in mc units e2 =_c ¼ 1=137:04 e2 =mc2 ¼ classical electron radius; r 0 m ¼ electron mass
y1 ¼ ffðk; p1 Þ
w ¼ ffðp2 ; p1 Þ y1c ¼ 1=1 photon characteristic angle wc ¼ y1c k=2 electron characteristic angle (1) We then have 2 p2 ¼ 1.
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
269
3. General properties of bremsstrahlung and the functioning of a bremsstrahlung tagger In the electron-bremsstrahlung process, an electron with energy 1, when passing through the Coulomb field of a nucleus, radiates a photon with energy k. This energy k can take any value up to the kinetic energy of the electron. Since the photon and scattered electron are emitted in the forward direction and the mass of the electron is small compared to that of the nucleus, the momentum transferred to the nucleus is small and the recoil energy given to the nucleus can be neglected. The energy loss of the electron corresponds, therefore, to the energy of the bremsstrahlung photon. In the tagging technique, the energy of the post-bremsstrahlung electron, 2 , is measured in a wide band magnetic spectrometer, the tagger, and the energy of the tagged photon can be determined as k ¼ 1 2 . Typically the tagger is provided with a well granulated focal plane detector, such that 2 can be measured with the desired precision. In order to have a beam spot with a defined, limited size, the photons have to be collimated before they hit the target of interest. The number of electrons detected in the spectrometer is always larger than the number of coincident photons that hit the target under investigation; this is due not only to the photon collimation but also to the fact that electrons from radiationless reactions are detected in the spectrometer (e.g., Møller scattering). However, the number of electrons is always proportional to the number of tagged photons that pass the collimator. The ratio of the number of photons coincident with an electron (the tagged photons) to the total number of detected electrons is called the tagging efficiency Z. This quantity can easily be measured, e.g.. when, instead of the target of interest, a photon detector with known efficiency is installed in the photon beam and the tagged photons are scaled in coincidence with the number of electrons. Knowing Z one obtains the photon flux on the target of interest from the flux of detected electrons. A typical value of the tagging efficiency, 50%, is found when the photon collimation allows only photons with emission angles smaller than 1 characteristic angle, y1c ¼ 1=1 , to pass the collimator and all electrons are detected. The tagged photon flux is, in this technique, limited by the fact that for all tagged photons, the corresponding electrons have to be detected independently of whether the photon will cause a reaction in the target of interest or not. In order to increase the possible flux of photons, the granulation of the focal plane electron detector can be enhanced but there are limits to this technique, since the counting capabilities of single electron detectors have to be high. Another way to increase the usable photon flux is to avoid detecting electrons that have no photon partner passing the collimator. This can be done by making use of the angular correlation between the electron and the photon. Since the momentum transferred to the nucleus in the bremsstrahlung process is small, a substantial correlation is to be expected. The bremsstrahlung cross-section is large for small momentum transfers, q. The momentum transferred to the nucleus in the process of interest is an important parameter that influences the electron–photon angular correlation. Therefore, in our calculations screening corrections have to be taken into account since they decrease the bremsstrahlung cross-section significantly in the region of small momentum transfers. Since this is the only place where Z plays a role for the features discussed here, we always show cross-sections divided by Z 2. Bremsstrahlung crosssections scale well when polar angles are expressed in characteristic angles for the photon and the post-bremsstrahlung electron. So, in the figures that follow, the angles y1 and w are given in units of the respective characteristic angles.
radiator
electron detector
Fig. 1. The figure shows the optics of a typical tagger. Electrons with different polar angles start from the radiator. The dotted lines are electrons with polar angles in the bend plane, in which there is point to point focusing. Therefore, the main trajectory (solid line) and the dotted lines meet at the electron detector. The dashed lines show electrons with polar angles perpendicular to the bend plane. As shown in the figure, there is no focusing for these electrons. Thus, when they have large enough polar angles they miss the electron detector. If the bend plane is coincident with the x-plane, then the x-component of the electron polar angle gets focused, while electrons with a y-component larger than a corresponding width of the electron detector, miss this detector, and can said to be ‘‘collimated’’.
Tagging spectrometers are always built to have a good momentum resolution for the post-bremsstrahlung electron. This requires that in the bend plane point to point optics are realized, which leads to an integration over the angular distribution of the electrons in this plane. In the plane perpendicular to the bend plane one has, however, the chance to avoid the detection of electrons with large angles, for example by limiting the size of the electron detectors in this direction. The general optics of a tagging spectrometer are shown in Fig. 1.
4. The Bethe–Heitler cross-section In this section, we give the principal equations in the derivation of the angular distribution of scattered electrons associated with collimated bremsstrahlung from high energy electrons. The starting point for our derivation is the Bethe–Heitler (first Born approximation) cross-section. We are primarily interested in small angles between the momenta of the incident electron, the scattered electron, and the emitted photon, p1 ; p2 , and k, respectively, in which case the momentum transfer, q, is of order mc ¼ 1 or smaller. We, therefore, neglect the effects of nuclear recoil but include the effects of atomic screening. However, we make no high energy or small angle approximations in the analysis of the Bethe–Heitler cross-section. As generally written, the Bethe–Heitler cross-section [1], is expressed in a coordinate system in which the z-axis is in the direction of the emitted photon, k. In this system, the direction of the incident beam is given by the angles ðy1 ; f1 Þ and the direction of the scattered electron is given by ðy2 ; f2 Þ, where
y1 ¼ ffðp1 ; kÞ y2 ¼ ffðp2 ; kÞ
(2)
The azimuthal angles f1 and f2 are in the plane perpendicular to k. With the experimental setup in mind, we choose a coordinate system with the z-axis in the direction of the incident electron beam, p1 , rather than the direction of the emitted photon, We, therefore, express the cross-section in terms of the angles w; jp for the scattered electron, and y1 ; jk for the photon. Here
w ¼ ffðp2 ; p1 Þ y1 ¼ ffðk; p1 Þ
(3)
and the azimuthal angles jp and jk are in the plane perpendicular to p1 .
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The needed relations between these variables in the two coordinate systems are cos y2 ¼ cos y1 cos w þ sin y1 sin w cosðjk jp Þ
(4)
and cos w ¼ cos y1 cos y2 þ sin y1 sin y2 cosðf1 f2 Þ
(5)
from which sin y1 sin y2 cosðf1 f2 Þ ¼ cos wsin2 y1 cos y1 sin y1 sin w cosðjk jp Þ
(6)
The momentum transfer is related to the other variables by
with
a1 ¼ 0:10;
a2 ¼ 0:55;
bi ¼ ðZ 1=3 =121Þbi ;
a3 ¼ 0:35
b1 ¼ 6:0;
b2 ¼ 1:20;
b3 ¼ 0:30
(16)
Figs. 2 and 3 show correlations between the photon and the electron polar angles, y1 and w, for 1 ¼ 12 GeV and k ¼ 6 GeV and a difference in the azimuth of p. The sharp dips when the polar angles in units of the respective characteristic angles are equal result from phase space (see also Figs. 5–7 in Ref. [4]).
5. Integration over the azimuth
q ¼ ðp1 p2 kÞ2 2
2
2
¼ ðp1 p2 Þ2 k þ 2k 2kðp1 p2 Þ 2
¼ 2kd1 2kd2 þ m
(7)
where d1 ¼ ð1 p1 cos y1 Þ d2 ¼ ð2 p2 cos y2 Þ
(8)
m2 ðp1 p2 Þ2 k2 ¼ 2ð1 2 p1 p2 cos w 1Þ m20 ðp1 p2 Þ2 k2 ¼ 2ð1 2 p1 p2 1Þ
(9)
We start with the totally differential Bethe–Heitler crosssection for the scattering of an electron with the emission of a photon 2 e2 e2 ½1 FðqÞ2 p2 dk dOk dOp2 3 f gBH d s ¼ Z2 _c mc2 p1 k ð2pÞ2 q4
(10)
where f gBH ¼
With the integration over jk in mind, we note from Eqs. (4) and (7) that in the above expression for the cross-section the variables that depend on jk are d2 and q2 . Writing f ¼ jk jp , the integral over the photon azimuthal angle is then, from Eqs. (13) and (15), 2 e2 e2 p2 dk sin y1 dy1 dOp2 3 d s ¼ Z2 2 _c mc p1 k ð2pÞ2 9 8 Z 2p Z 2p 3 = < X X ai aj 2 2 ½ ij df þ ai ½ i df 2 2 ; : 0 ðb b Þ 0 1piojp3
in which " ½ ij ¼
2
2 2
2
sin y1 ð4 q Þ
ð1 p1 cos y1 Þ2
þ
p22
2
2 1
2
sin y2 ð4 q Þ
2
2
q2 þ bi
2
q2 þ bj
1 2
ðq2 þ bi Þ2
(20)
2
1 = 12 GeV, k = 6 GeV, k - p =
p1 p2 sin y1 sin y2 cosðf1 f2 Þ ¼ p1 p2 cos w ð1 d1 Þð2 d2 Þ (12) The expression f gBH in Eq. (11) may then be written as ( 2 1 2 2ð21 þ 22 Þ m2 2 f gBH ¼ ð2kÞ2 4 þ q 1
D2
D1 D2
1 1 1 1 1 þ q2 þ ðq2 m2 Þ D1 D2 D1 D2 2
2 )
(13)
where
5 2
1/c = 0.0
Z=6
1/c = 0.5 1/c = 1.0
12
d3 /(Z2 dk dΩk dΩp2)/(barns/(sr2 mc2))
Eqs. (4), (6) and (7) then express all of the terms in Eq. (11) in the variables of the p1 system. In view of the denominators ð1 p1 cos y1 Þ and ð2 p2 cos y2 Þ, we write Eq. (11) in the variables given in Eqs. (8) and (9). From Eq. (5) we then have
10
5
1/c = 1.5 Z = 79
2 1011 5 2 1010 5 2 109
(14)
For the atomic form factor in Eq. (10) we use the expression given by Molie`re [2], viz., 3 1 FðqÞ X ai ¼ 2 2 q2 i¼1 q þ bi
(19)
D2 ¼ 2kd2 ¼ c þ d cos f
(11)
D1 ¼ 2kd1 D2 ¼ 2kd2
(18)
f gBH
f gBH
2k ðp21 sin y1 þ p22 sin y2 2p1 p2 sin y1 sin y2 cosðf1 f2 ÞÞ ð1 p1 cos y1 Þð2 p2 cos y2 Þ
D1
#
1
From Eq. (4) we have
ð2 p2 cos y2 Þ2
2p p sin y1 sin y2 cosðf1 f2 Þð41 2 q2 Þ 1 2 ð1 p1 cos y1 Þð2 p2 cos y2 Þ þ
2
(17)
i¼1
i
and ½ i ¼
p21
1
j
(15)
5 -1.0
-0.5
0.0
0.5 /c
1.0
1.5
2.0
Fig. 2. The Bethe–Heitler cross-section with screening correction as a function of the electron polar angle for different fixed photon polar angles and different values of Z.
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
271
where (
2
422 þ bi
X i ¼ ð2kÞ2
D21
!
2
I10;i 2
41 2 þ bi
D1
2
I11;i þ ð421 þ bi ÞI12;i
1 1 2 I10;i ðm2 þ bi Þ I10;i I11;i D1 2 2 2 2 2 2ð1 þ 2 Þ m I11;i bi
(26)
D1
d3
(
2
422 þ bi
Y i ¼ ð2kÞ2
D21
1
c
1.1
2
3
1
4
5
grees
p / de
Fig. 3. The Bethe–Heitler cross-section with screening correction as a function of electron polar angle and the electron azimuth. The photon polar angle is y1c . Note that the line for jp ¼ 0 corresponds to the curve in Fig. 2 for y1 =y1c ¼ 1 and Z ¼ 6.
in which
2 e2 e2 p2 dk sin y1 dy1 dOp2 3 d s ¼ Z2 _c mc2 p1 k ð2pÞ2 9 8 = < 3 X X ai aj 2 ½X 2 X þ a Y i j i i 2 2 ; : ðb b Þ
c ¼ 2kð2 p2 cos y1 cos wÞ (21)
c d ¼ 2kð2 p2 cosðy1 wÞÞ40
I01 ¼
The variables ai and b are defined by
I20;i
2
q2 þ bi ¼ ai þ b cos f
(22) I02
from which, using Eqs. (4) and (7), 2
ai ¼ 2kd1 þ m2 2kð2 p2 cos y1 cos wÞ þ bi b ¼ 2kp2 sin y1 sin w ¼ d
I11;i (23)
from which, with some algebra, I21;i
2
ai b ¼ d þ bi þ 4dk sin2 12 y1 þ 4dp2 sin2 12w þ 4ðp2 þ kÞ2 sin2 12 y1 sin2 12 w þ 4ðp2 sin 12 w cos 12y1 k cos 12w sin 12y1 Þ2 40
I12;i
The integrals over f are, thus, all of the form df
0
ðq2 þ bi Þm Dn2
Z 2p ¼ 0
2
df ðai þ b cos fÞm ðc þ d cos fÞn
I22;i m; n ¼ 0; 1; 2
(24)
From Eqs. (18)–(24) the integrals over the photon azimuthal angle in Eq. (17) can then be written in the form
0
0
½ ij df ¼ X i X j ½ i df ¼ Y i
i
(28)
i¼1
I00 ¼ 2p I10;i ¼
from which
Z 2p
j
All of the integrals Imn;i can be calculated recursively from Im0;i and I0n . From Eq. (24) we have
d ¼ 2kp2 sin y1 sin w
2
(27)
The cross-section integrated over the photon azimuthal angle, given in Eq. (17), is thus expressed in terms of X i and Y i by Eqs. (25)–(27) in terms of the integrals Imn;i defined in Eq. (24), specifically,
1piojp3
Z 2p
D1
2
I21;i þ ð421 þ bi ÞI22;i
D1
D1
/
Z 2p
41 2 þ bi
2 2ð1 þ 22 Þ m2 2 I20;i þ I11;i bi I21;i D1 1 1 1 1 2 I20;i I21;i þ I10;i I11;i ðm2 þ bi Þ D1 2 2 D1 !) 1 2 þ I I11;i þ I12;i 2 10;i
0 0.9
Imn;i
!
2
I20;i 2
(25)
2p ða2i
2
b Þ1=2 2p 2
ðc2 d Þ1=2 @ 2pai ¼ I ¼ @ai 10;i ða2 b2 Þ3=2 i @ 2pc ¼ ; I01 ¼ 2 @c ðc2 d Þ3=2 Z 2p 1 1 1 df ¼ þ c þ ai 0 ai þ b cos f c þ d cos f 1 ¼ ðI þ I01 Þ c þ ai 10;i @ ¼ I @ai 11;i 1 ¼ ðI þ I11;i Þ c þ ai 20;i @ ¼ I @c 11;i 1 ¼ ðI þ I02 Þ c þ ai 11;i @ ¼ I @ai 12;i 1 ¼ ðI þ I12;i Þ c þ ai 21;i
(29)
Here 2
c þ ai ¼ D1 þ m2 þ bi G1;i
(30)
The variables needed for the computation of the integrals Imn;i 2 2 are ai ; c; c þ ai ; a2i b , and c2 d . We next integrate the crosssection over the photon polar angle corresponding to the collimator. However, for the integration we choose the variable
ARTICLE IN PRESS 272
L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
D1 rather than y1 , defined in Eqs. (8) and (14) by D1 ¼ 2kd1 ¼ 2kð1 p1 cos y1 Þ, from which dD1 ¼ 2kp1 sin y1 dy1 . We integrate the cross-section over y1 from y1 ¼ 0 to y0 and note that for y1 ¼ 0 we have D1 ¼ D1;0 ¼ 2kð1 p1 Þ ¼ 2k=ð1 þ p1 Þ and for y1 ¼ y0 we have D1 ¼ D1;1 ¼ 2kð1 p1 cos y0 Þ ¼ 2k=ð1 þ p1 Þþ 2kp1 ð1 cos y0 Þ ¼ 2k=ð1 þ p1 Þ þ 4kp1 sin2 12y0 . We then have 2 e2 e2 p2 dk dOp2 1 _c mc2 p1 k ð2pÞ2 2p1 k 8 Z D1;1 < X ai aj 2 ½X i X j dD1 2 2 : 1piojp3 ðbj bi Þ D1;0 ) Z D1;1 3 X 2 þ ai Y i dD1
2
Z
D1;1
D1;0
Y i dD1 ¼ ð2kÞ2
(31)
D1;0
2
2
þ ð2ð21 þ 22 Þ m2 ÞðJ 1;11;i bi J1;21;i Þ 1 2 ðm2 þ bi ÞðJ1;20;i J0;21;i Þ 2 1 þ ðJ 1;10;i J 0;11;i Þ þ ðJ2;10;i 2J1;11;i þ J 0;12;i Þ 2
D1;0
Imn;i
Dl1
dD1
(32)
D1;1 D1;0
2
X i dD1 ¼ ð2kÞ
l ¼ 0; 1
2 2
ðð4 þ
b2i ÞJ 2;10;i
2ð41 2 þ
(35)
J0;2n;i
n ¼ 0; 1; 2
J1;2n;i
n ¼ 0; 1
I12;i ¼
2
I22;i ¼
2
bi ð2ð21 þ 22 Þ m2 ÞJ 1;11;i
(33)
(36)
Using the recursion relations (29) for Imn;i, we need only to calculate explicitly the integrals for which either m or n is zero, the remaining integrals then being expressed in terms of Im0;i and I0n;i . To this end, we first note, from Eqs. (29) and (30) that
I21;i ¼
b2i ÞJ 1;11;i
þ ð421 þ bi ÞJ 0;12;i Þ J 0;10;i 1 2 ðm2 þ bi ÞðJ 1;10;i J 0;11;i Þ 2
1
G1;i 1
G1;i 1
G1;i 1
G21;i
ðI10;i þ I01 Þ I20;i þ I02;i þ
1 = 12 GeV, k = 6 GeV Z=6
1/1c = 0.5
d3 /(Z2 dk d1 dΩp2)/(barns/(rad sr mc2))
10
13
5
Z = 79
ðI10;i þ I01 Þ 2
G31;i
ðI10;i þ I01 Þ
Defining Z D1;1 Imn;i K l;mn;i dD1 l
G1;i
1/1c = 1.0
J0;11;i ¼ K 1;10;i þ K 1;01;i
1/1c = 1.5
J0;12;i ¼ K 1;02;i þ K 2;10;i þ K 2;01;i
(37)
(38)
J0;21;i ¼ K 1;20;i þ K 2;10;i þ K 2;01;i
2
J0;22;i ¼ K 2;20;i þ K 2;02;i þ 2K 3;10;i þ 2K 3;01;i 1 J1;11;i ¼ ðJ 1;10;i þ J 1;01 K 1;10;i K 1;01;i Þ m2 þ b2i 1 ðJ 1;02 K 1;02;i K 2;10;i K 2;01;i Þ J1;12;i ¼ m2 þ b2i 1 ðJ 1;10;i þ J1;01 K 1;10;i K 1;01;i Þ þ 2 ðm2 þ bi Þ2 1 ðJ 1;20;i K 1;20;i K 2;10;i K 2;01;i Þ J1;21;i ¼ m2 þ b2i 1 ðJ 1;10;i þ J1;01 K 1;10;i K 1;01;i Þ þ 2 ðm2 þ bi Þ2
1012 5 2 1011 5 2 1010 5 2 109 5 -1.0
1
G21;i
ðI10;i þ I01 Þ
we can then express all of the integrals in Eqs. (33) and (34) in terms of integrals J l;mn;i and K l;mn;i in which either m or n is zero. From Eqs. (32), (37) and (38) we have
1/1c = 0.0
2
1
G21;i
ðI20;i þ I02 Þ þ
D1;0
5
(34)
and the six in which m ¼ 2, namely
I11;i ¼
From Eqs. (26) and (27) Z
l ¼ 0; 1; 2
J2;20;i
Substituting Eqs. (26) and (27) in Eq. (28), the integrals over the photon polar angle are all of the form D1;1
Jl;10;i ; Jl;11;i ; J0;12;i
6. Integration over the photon polar angle
Z
2
As can be seen in the above expressions, the needed integrals are the six in which m ¼ 1, namely
In Fig. 4 a resulting cross-section is shown, which corresponds to Fig. 2 before integration.
J l;mn;i
2
ðð422 þ bi ÞJ 2;20;i 2ð41 2 þ bi ÞJ1;21;i
þ ð421 þ bi ÞJ 0;22;i Þ J0;20;i
d s ¼ Z2
i¼1
-0.5
0.0
0.5 / c
1.0
1.5
2.0
Fig. 4. The Bethe–Heitler cross-section with screening correction integrated over the photon azimuth jk .
(39)
The required integrals are now Jl;10;i ;
l ¼ 0; 1; 2
Jl;20;i ;
l ¼ 0; 1; 2
J1;01
(40)
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
(
and K l;10;i ;
l ¼ 1; 2; 3
K l;01;i ;
l ¼ 1; 2; 3
K l;20;i ; K l;02;i ;
l ¼ 1; 2 l ¼ 1; 2
) D1;1
B ð2pÞ2
2;i K 1;10;i
B3;i G1;i I10;i
B3;i D1;0 ( ) D1;1
D ð2pÞ2
¼ 2;i K 1;01;i
D3;i G1;i I01
D3;i
K 2;10;i ¼
K 2;01;i (41)
We note that although ai ; c and c þ ai are linear functions of 2 2 cos y1 and that a2i b and c2 d are quadratic functions of cos y1 , the expressions are simplified if we use D1 ¼ 2kd1 rather than cos y1 as the integration variable. In view of the integrands in 2 2 Eqs. (32) and (38) it is useful to write c2 d and a2i b alternatively as quadratic functions of D1 and of G1;i . We then have, with a bit of algebra,
ai ¼ g1 D1 þ g2;i ¼ g1 G1;i þ g3;i
K 3;10;i ¼
K 3;01;i ¼
J 2;20;i ¼
(42)
and 2
c2 d ¼ C 1 D21 þ 2C 2 D1 þ C 3 ¼ D1 G21;i þ 2D2;i G1;i þ D3;i 2
a2i b ¼ A1 D21 þ 2A2;i D1 þ A3;i ¼ B1 G21;i þ 2B2;i G1;i þ B3;i
(43)
K 2;20;i ¼
where from Eqs. (21) and (23) p2 cos w p1 2k 1 d2 ¼ 2 k þ 1 m2 2 p1
d1 ¼
d3;i ¼ d2 d1 ðm2 þ b2i Þ
2 p2 p1 2k 1 C 2 ¼ 2 k 2 m2 2 p1 2 2k 1 2 2 C3 ¼ m ðm þ 4Þ p1 4
K 2;02;i ¼
C1 ¼
2
D2;i ¼ C 2 C 1 ðm2 þ bi Þ 2
2
D3;i ¼ C 3 2C 2 ðm2 þ bi Þ þ C 1 ðm2 þ bi Þ2
(44)
and
g1 ¼ 1 d1
D1;0
þ
1 2 2 ðm þ k Þ p21 1 1 2 1 2 ¼ 2 m2 m k2 þ b2i m þ k1 2 2 p1 1 1 2 2 2 2 ¼ 2 ðm p2 bi p1 Þ þ 8bi p1 p2 ð1 2 þ p1 p2 1Þsin2 w 2 p1
A3;i
þ
B3;i
K 1;02;i ¼ þ
The integrals with l ¼ 2 or 3 can be expressed in terms of integrals for which l ¼ 0 or 1
K 1;10;i
D þ d1 d3;i 2;i D3;i D3;i
(45)
D1;0
g3;i
g3;i
("
2
) D1;1
A ð2pÞ2
¼ 2;i J1;10;i
A3;i D1 I10;i
A3;i D1;0 ( ) D1;1 ð2pÞ2
C2 ¼ J
C 3 D1 I01
C 3 1;01
(47)
1;0
B2;i ¼ A2;i A1 ðm2 þ bi Þ
(
J 1;10;i
!# ) D1;1
B B1 G1;i þ B2;i
I þ g1 g3;i 2;i 10;i
2
B3;i B3;i B1 B3;i B2;i D
K 1;20;i ¼
B1 ¼ A 1 B3;i ¼ D3;i
g2;i
("
A1 ¼
J 2;01
(46)
Furthermore, the integrals with l ¼ 1 and either m ¼ 2 or n ¼ 2 can be expressed in terms of the integrals with l ¼ 1 and either m ¼ 1 or n ¼ 1 (" !# ) D1;1
g2;i A A1 D1 þ A2;i
I þ g1 g2;i 2;i J1;20;i ¼
10;i 2
A3;i A3;i A1 A3;i A2;i
g2;i ¼ d2 þ ðm2 þ b2i Þ g3;i ¼ d3;i
J 2;10;i
) D1;1
3B ð2pÞ B1
2;i K 2;10;i K
2 2B3;i 2B3;i 1;10;i 2B3;i G1;i I10;i D 1;0 ( ) D1;1
3D2;i ð2pÞ2 D1
K K
2 2D3;i 2;01;i 2D3;i 1;01;i 2D3;i G1;i I01 D 1;0 0(" 1 # ) D1;1
A2;i ðA1 D1 þ A2;i Þ g1 @
I10;i
1 þ J1;10;i A
A3;i A1 A3;i A22;i D1;0 (" ! g2;i 3A2;i ðA1 D1 þ A2;i Þ 1 þ þ A3;i D1 A3;i A1 A3;i A22;i 1 !# )
D1;1 3A 3A22;i
2;i A I10;i
þ J 2A1
A3;i A3;i 1;10;i D1;0 0(" 1 # ) D1;1
B2;i ðB1 G1;i þ B2;i Þ g1 @
I10;i
1 þ K 1;10;i A
B3;i B1 B3;i B22;i D1;0 (" ! g3;i 3B2;i ðB1 G1;i þ B2;i Þ 1 þ þ B3;i G1;i B3;i B1 B3;i B22;i 1 !# ) D1;1
3B22;i 3B2;i
A 2B1 I10;i
þ K
B3;i B3;i 1;10;i D1;0 0(" 1 # ) D1;1
D2;i ðD1 G1;i þ D2;i Þ d1 @
I01
1 þ K 1;01;i A
D3;i D1 D3;i D22;i D1;0 (" ! d 3D2;i ðD1 G1;i þ D2;i Þ 1 þ þ 3;i D3;i G1;i D3;i D1 D3;i D22;i 1 !# )
D
1;1 3D 3D22;i
2;i A 2D1 I01
þ K
D3;i D3;i 1;01;i 2
D1;0
D1 ¼ C 1
A3;i
D1;0
(
c ¼ d1 D1 þ d2 ¼ d1 G1;i þ d3;i
A2;i
273
d3;i
d3;i D3;i
K 1;01;i
(48)
D1 G1;i þ D2;i D1 D3;i D22;i
!#
) D1;1
I01
D1;0
(49)
We now have only to evaluate explicitly the integrals for which l ¼ 0 or 1, in terms of which we have expressed all the other integrals. These are ( ) D1;1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
J0;10;i ¼ 2p pffiffiffiffiffiffi lnð ðA1 D1 þ A2;i Þ2 þ A1 A3;i A22;i þ ðA1 D1 þ A2;i ÞÞ
A1 D 1;0
(50)
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
(" J 0;20;i ¼
g1 ðA2;i D1 þ A3;i Þ þ g2;i ðA1 D1 þ A2;i Þ A1 A3;i
A22;i
#
) D1;1
I10;i
0<1/1c<1
D1;0
(52)
J 1;01
8 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 D1;1
< 1 ðC 2 D1 þ C 3 Þ2 þ ðC 1 C 3 C 22 ÞD21 þ ðC 2 D1 þ C 3 Þ =
A
¼ 2p pffiffiffiffiffiffi ln@ ;
: C3 D1 D1;0
(53)
K 1;10;i
0.009
D1;0
8 19 D1;1 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
< 1 ðA2;i D1 þ A3;i Þ2 þ ðA1 A3;i A22;i ÞD21 þ ðA2;i D1 þ A3;i Þ =
A
¼ 2p pffiffiffiffiffiffiffi ln@ : A3;i ;
D1
J1;10;i
1 = 12 GeV, k = 6 GeV, Z = 6, int 1
(51)
8 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 D1;1 < 1 ðB2 G1;i þ B3;i Þ2 þ ðB1 B3;i B22;i ÞG21;i þ ðB2 G1;i þ B3;i Þ =
A
¼ 2p pffiffiffiffiffiffiffi ln@ : B3;i ;
G1;i D1;0
1<1/1c<2
0.008
2<1/1c<3
d2 /(Z2 dk dΩp)/(barns/(sr mc2))
274
(54)
0.007
0<1/1c<10
0.006 0.005 0.004 0.003 0.002 0.001
8 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 19 D1;1 < 1 ðD2;i G1;i þ D3;i Þ2 þ ðD1 D3;i D22;i ÞG21;i þ ðD2 G1;i þ D3;i Þ =
A
K 1;01;i ¼ 2p pffiffiffiffiffiffiffiffi ln@ : D3;i ;
G1;i
0.0
D1;0
1
0
(55)
2
3 / c
Here 2
2 2
A1 A3;i A22;i ¼ B1 B3;i B22;i ¼ ½ðm2 þ bi Þ2 þ 4bi k
p22 sin2 w p21
(56)
4
5
6
Fig. 5. Angular distribution of post-bremsstrahlung electrons after integration over different intervals of the photon polar angle y1 .
and p22 sin2 w p21
0.009 (57)
Fig. 5 shows the angular distribution of the post-bremsstrahlung electrons as a function of the polar angle of the scattered electron for different photon collimations. The angular correlation can be seen clearly.
7. Final results and conclusion In order to investigate the angular correlation between a bremsstrahlung photon and its corresponding post-bremsstrahlung electron within the context of a tagging spectrometer, we integrated analytically the Bethe–Heitler (first Born approximation) cross-section, including the Molie`re screening correction. Our results are given in terms of angular distributions of the postbremsstrahlung electron associated with photons that pass through a circular collimator centered in the forward direction. To draw conclusions from this distribution we assume optical properties of the magnetic tagging spectrometer as indicated in Fig. 1. In the bend plane of the spectrometer, we have point to point optics in order to achieve good momentum resolution. This results in integration over the respective component of the electron polar angle. However, for the component perpendicular to the bend plane, we have the freedom to assume optics in which the electron polar angle results in a position that can be collimated by the size of the electron detectors. Integration over the angular distributions in Fig. 6 gives the cross-section for the different combinations of collimation, denoted by sij, with i and j representing the photon and the electron collimation, respectively, where 0 stands for off and 1 for on. We can then determine ratios of the si;j , which stand for the following efficiencies. The tagging efficiency Z is defined as the number of tagged photons through the collimator divided by the total number of detected electrons, collimated or uncollimated. For uncollimated electrons we find the efficiency Z0 s10 =s00. For the collimated
0.008
d2 /(Z2 dk dΩp)/(barns/(sr mc2))
C 1 C 3 C 22 ¼ D1 D3;i D22;i ¼ ð2kÞ2
0.007 0.006
1 coll
coll
off
off
on
off
off
on
on
on
0.005 0.004 0.003 0.002 0.001 0.0 0
1
2
3
4
5 /c
6
7
8
9
10
Fig. 6. Angular distribution of the electrons in coincidence with photons that pass the collimator, for different collimation of the photons and the electrons. There is either no collimation (off) or a collimation to 1 characteristic angle (on). All combinations are shown.
case we have Z s11 =s01 . The photon probability, b, is given by the ratio of tagged photons to all photons through the collimator, i.e. b s11 =s10 . Finally, we define l, which is the ratio of collimated electrons to uncollimated electrons, l s01 =s00 , which gives the reduction of electron count rate due to electron collimation. The above defined ratios are shown as a function of w=wc in Fig. 7. The tagging efficiency for uncollimated electrons, Z0 , is of course constant. Once the electrons are collimated, the tagging efficiency Z increases with decreasing number of detected
ARTICLE IN PRESS L.C. Maximon et al. / Nuclear Instruments and Methods in Physics Research A 603 (2009) 268–275
275
1.0 0.9 0.8
efficiencies
0.7 0.6 0.5
Z=6
Z = 79
0.4
1 coll = 0.5 1c
1 coll = 0.5 1c
0.3 0.2 0.1 0.0
0.9 0.8
efficiencies
0.7 0.6 0.5 0.4 0.3
Z=6
Z = 79
0
0.2
1 coll = 1.0 1c
1 coll = 1.0 1c
0.1
/
0.0 0.0
0.5
1.0 1.5 2.0 2.5 electron collimation in c
0.0
0.5
1.0 1.5 2.0 2.5 electron collimation in c
3.0
Fig. 7. Efficiencies deduced from the values of the sij for different photon collimations and Z. See text.
electrons, while the photon probability b decreases (since now we have photons through the collimator that are not tagged). In a tagged photon experiment, the number of ‘‘good’’ reactions is proportional to the number of tagged photons. When the postbremsstrahlung electrons are collimated, the photon probability b is no longer equal to one and there is a reduction by a factor of b in the number of tagged photons. To compensate for this reduction in tagged photons, the incoming electron current can be increased by a factor of 1=b so as to keep the number of ‘‘good’’ reactions the same as when there is no collimation on the post-bremsstrahlung electrons. When the post-bremsstrahlung electrons are collimated and the incoming electron current is increased by 1=b, the number of collimated electrons is decreased by a factor of l=b from the case where the post-bremsstrahlung electrons are not collimated, while the number of ‘‘good’’ events remains the same for each
case. Therefore, for a given good reaction rate, the tagger rate is decreased by a factor of l=b when the post-bremsstrahlung electrons are collimated.
Acknowledgment The work done at Arizona State University was supported by the National Science Foundation award PHY-0653630. References [1] [2] [3] [4]
H.W. Koch, J.W. Motz, Rev. Mod. Phys. 31 (1959) 920. G. Molie`re, Z. Naturforsch. 2a (1947) 133. D.H. Bailey, ACM Trans. Math. Software 19 (3) (1993) 288. L.C. Maximon, A. de Miniac, T. Aniel, Phys. Rep. 147 (1987) 189.